Method and apparatus for equal energy codebooks for antenna arrays with mutual coupling

ABSTRACT

A method and apparatus provide equal energy codebooks for antenna arrays with mutual coupling. A plurality of precoders can be received from a codebook in a transmitter having an antenna array. Each precoder of the plurality of precoders can be transformed using a transformation that maps each precoder to equal radiated power to generate transformed precoders. A transformation matrix for the transformation can be a matrix with column vectors equal to the eigenvectors of a Hermitian and non-negative definite matrix multiplied by a diagonal matrix with the value of each diagonal element equal to the inverse of the positive square root of the eigenvalue of the corresponding eigenvector. A signal can be received for transmission. A transformed precoder of the plurality of transformed precoders can be applied to the signal to generate a precoded signal for transmission over a physical channel. The precoded signal can be transmitted.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of an application entitled“Method and Apparatus for Equal Energy Codebooks for Antenna Arrays withMutual Coupling,” U.S. application Ser. No. 14/855,693, filed on Sep.16, 2015, and commonly assigned to the assignee of the presentapplication, which is hereby incorporated by reference in its entirety.

BACKGROUND

1. Field

The present disclosure is directed to a method and apparatus for equalenergy codebooks for antenna arrays with mutual coupling.

2. Introduction

Presently, wireless communication devices communicate with othercommunication devices using wireless signals. Many wirelesscommunication devices have multiple antennas that can transmit morefocused signals to a receiving device using antenna beamforming.Multiple telecommunication standards define antenna precoder codebooksto support antenna beamforming and or multiple-input/multiple output(MIMO) transmission with feedback from a receiver in the receivingdevice. Precoders from the codebooks apply a vector of complexamplitudes to a transmitted signal to provide for the antennabeamforming. Telecommunication standards that employ codebooks ofprecoders include the 3GPP HSPA and LTE standards and the IEEE 802.11and 802.16 standards. In all of these standards, the precoders that aredefined have the property that each precoding vector has equal L² normwith the assumption that the precoders are applied to the antenna arrayin such a way that precoders having equal L² norm yield antenna patternswith equal radiated power in the far field. Here the L² norm (also knownas the Euclidean norm) is defined as the square root of the sum of thesquare of the amplitudes so that for a vector x of length Mx=(x ₁ x ₂ . . . x _(M))^(T),the L² or Euclidean norm is given by

$\left( {\sum\limits_{i = 1}^{M}{x_{i}}^{2}} \right)^{\frac{1}{2}}$

In each of the above telecommunication standards, precoders are used incombination with reference symbol transmissions so that the receiver canestimate and evaluate the quality of the channel that would result fromapplication of each of the precoders. The receiver applies each of theprecoders to the reference symbols in order to evaluate the channelquality. It then signals the index of the best precoder and thecorresponding channel quality back to the transmitter. For sometransmission modes, the precoder used for the data transmission issignaled from the transmitter to the receiver, and the receiver thenapplies the precoder to the channel estimates of the reference symbolsin order to estimate the channel for the data symbols.

Implicit in the operation of these types of systems is the assumptionthat the precoders are applied in a manner such that the antenna patterncorresponding to each precoder has the same radiated power in the farfield. The reason for this assumption is that it is the objective of thereceiver to select the precoder which maximizes the channel quality, andthus the achievable data rate, for a given transmit power. In the caseof a single user, this will maximize the transmission range of a fixeddata rate, or alternatively, the achievable data rate at a fixed range.Alternatively, for multi-user systems, it is desirable to minimize thetransmit power needed to achieve a given data rate for each user, as thetransmit power for the target user is interference for all users otherthan the target user.

If there is no mutual coupling of the transmit array, then it will betrue that antenna precoding vectors having equal L² norm will yieldantenna patterns with equal power (some assumptions are necessary; e.g.,such as the source impedances are equal and the antenna elements haveequal self-impedance). However, if the antennas are coupled, then theantenna patterns resulting from two precoders having the same L² normcan differ in transmit power by several dB. The amount of thisdifference depends on multiple factors, including the mutual couplingcoefficients, the type of sources used to drive the array, and thesource impedances. Unfortunately, this difference results in a receivermaking errors when evaluating channel quality from antenna arrays withmutual coupling.

Thus, there is a need for a method and apparatus for equal energycodebooks for antenna arrays with mutual coupling.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the manner in which advantages and features of thedisclosure can be obtained, a description of the disclosure is renderedby reference to specific embodiments thereof which are illustrated inthe appended drawings. These drawings depict only example embodiments ofthe disclosure and are not therefore to be considered to be limiting ofits scope.

FIG. 1 is an example block diagram of a system according to a possibleembodiment;

FIG. 2 is an example illustration of a two-port model for a two-elementarray according to a possible embodiment;

FIG. 3 is an example illustration of a Thevenin source model accordingto a possible embodiment;

FIG. 4 is an example illustration of a Norton source model according toa possible embodiment;

FIG. 5 is an example graph illustrating transmitter power vs. relativephase offset for a Thevenin source model according to a possibleembodiment

FIG. 6 is an example graph illustrating transmitter power vs. relativephase offset for a Norton source model according to a possibleembodiment;

FIG. 7 is an example flowchart illustrating the operation of a wirelesscommunication device according to a possible embodiment; and

FIG. 8 is an example block diagram of an apparatus according to apossible embodiment.

DETAILED DESCRIPTION

Embodiments provide a method and apparatus for equal energy codebooksfor antenna arrays with mutual coupling. A plurality of precoders can bereceived from a codebook in a transmitter having an antenna array. Eachprecoder of the plurality of precoders can be transformed using atransformation that maps each precoder to a transformed precoder suchthe transformed precoders generate antenna patterns with equal radiatedpower. A transformation matrix for the transformation can be a matrixwith column vectors equal to the eigenvectors of a Hermitian andnon-negative definite matrix multiplied by a diagonal matrix with thevalue of each diagonal element equal to the inverse of the positivesquare root of the eigenvalue of the corresponding eigenvector. A signalcan be received for transmission. A transformed precoder of theplurality of transformed precoders can be applied to the signal togenerate a precoded signal for transmission over a physical channel. Theprecoded signal can be transmitted.

FIG. 1 is an example block diagram of a system 100 according to apossible embodiment. The system 100 can include a transmitting device110 and a receiving device 120. The transmitting device 110 can be aUser Equipment (UE), a base station, or any other device that cantransmit wireless signals. Similarly, the receiving device 120 can be aUE, a base station, or any other device that can received wirelesssignals. A UE can be a wireless terminal, a portable wirelesscommunication device, a smartphone, a cellular telephone, a flip phone,a personal digital assistant, a device having a subscriber identitymodule, a personal computer, a selective call receiver, a tabletcomputer, a laptop computer, or any other device that is capable ofsending and receiving wireless communication signals.

The transmitting device 110 can include a precoder transformationcontroller 112, a codebook 114, and an antenna array 116. The precodertransformation controller 112 can be one element or can be distributedbetween different elements. For example, the precoder transformationcontroller 112 can be part of a processor, can be part of a transceiver,can be part of a precoder, can be part of other elements in atransmitting device, and/or can be distributed between combinations ofelements in a transmitting device and/or over cloud computing. Thereceiving device 120 can include at least one antenna 122. For example,in some embodiments the receiving device 120 can have one antenna and inother embodiments the receiving device 120 can have an array ofantennas.

In operation, the precoder transformation controller 112 can receive aplurality of precoders from the codebook 114 in the transmitting device110 having the antenna array 116. The precoder transformation controller112 can transform each precoder of the plurality of precoders using atransformation that maps each precoder to a transformed precoder suchthat the transformed precoders generate antenna patterns with equalradiated power. The precoder transformation controller 112 can receive asignal 118 for transmission. The precoder transformation controller 112can apply a transformed precoder of the plurality of transformedprecoders to the signal 118 to generate a precoded signal fortransmission over a physical channel. The precoder transformationcontroller 112 can then transmit the precoded signal to the receivingdevice 120.

FIG. 2 is an example illustration of a two-port model 200 for atwo-element array according to a possible embodiment. The two elementscan correspond to two antennas in an antenna array. To determine thetransformation, an M-port circuit can be used to model the vectorvoltage-current relationship for the M-ports of an M-element antennaarray, which is given byv=Zi,where Z is the M×M impedance matrix for the array. In the two-port model200, i₁ and v₁ denote the current and voltage for a first antenna, whilei₂ and v₂ denote the current and voltage for a second antenna, and

$v = {{\begin{bmatrix}v_{1} \\v_{2}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu}{\mathbb{i}}} = {\begin{bmatrix}{\mathbb{i}}_{1} \\{\mathbb{i}}_{2}\end{bmatrix}.}}$

FIG. 3 is an example illustration of a Thevenin source model 300according to a possible embodiment. FIG. 4 is an example illustration ofa Norton source model 400 according to a possible embodiment. The twolinear source models can be considered for driving the antenna array. Ingeneral, the circuits used to drive each element of the antenna arraycan be modeled as a Thevenin source or as a Norton source. The Theveninsource model is used in the case that antenna precoders are applied asvoltages, and the Norton source is applied in the case that the antennaprecoders are applied as current sources. The Thevenin source caninclude an ideal vector voltage source v_(S) in combination with aseries impedance Z_(S) _(_) _(Thev), where Z_(S) _(_) _(Thev) is adiagonal matrix with diagonal elements equal to the series impedance foreach voltage source. The Norton source can include an ideal vectorcurrent source i_(S) in combination with a parallel shunt impedanceZ_(S) _(_) _(Nor) where Z_(S) _(_) _(Nor) is a diagonal matrix withdiagonal elements equal to the shunt impedance for each current source.It can be noted that the Norton source will yield two-port currents iand voltages v which are equal to that for the Thevenin source so longasZ _(S) _(_) _(Thev) =Z _(S) _(_) _(Nor) and v _(S) =Z _(S) _(_) _(Thev)i _(S)where

$v_{S} = {{\begin{bmatrix}v_{S\; 1} \\v_{S\; 2}\end{bmatrix}\mspace{14mu} Z_{S\;\_\;{Thev}}} = \begin{bmatrix}Z_{{S\;\_\;{Thev}},1} & 0 \\0 & Z_{{S\;\_\;{Thev}},2}\end{bmatrix}}$and

${\mathbb{i}}_{S} = {{\begin{bmatrix}{\mathbb{i}}_{S\; 1} \\{\mathbb{i}}_{S\; 2}\end{bmatrix}\mspace{14mu} Z_{S\;\_\;{Nor}}} = {\begin{bmatrix}Z_{{S\;\_\;{Nor}},1} & 0 \\0 & Z_{{S\;\_\;{Nor}},2}\end{bmatrix}.}}$

For an M-element array, the peak radiated power (average power isone-half of peak) is equal to the power delivered to the M-port deviceand is given byRe(v ^(H) i)=Re(i ^(H) Z ^(H) i)where Z is the impedance matrix and i is the vector of input currents.

For the Thevenin source model 300 with source voltage v_(S) and sourceimpedance Z_(S) _(_) _(Thev), the current vector at the input to thetwo-port device is given byi=(Z _(S) _(_) _(Thev) +Z)⁻¹ v _(S).

Thus, the radiated power for the Thevenin source is given byP _(rad) _(_) _(Thev)(v _(S) ,Z _(S) _(_) _(Thev) ,Z)=Re(((Z _(S) _(_)_(Thev) +Z)⁻¹ v _(S))^(H) Z((Z _(S) _(_) _(Thev) +Z)⁻¹ v _(S))).=Re(v _(S) ^(H)(Z _(S) _(_) _(Thev) +Z)^(−H) Z(Z _(S) _(_) _(Thev) +Z)⁻¹v _(S))

This expression can be further simplified as

$\begin{matrix}{{P_{{ra}\; d\;\_\;{Thev}}\left( {v_{s},Z_{S\;\_\;{Thev}},Z} \right)} = {\frac{1}{2}\left( {{{v_{S}^{H}\left( {Z_{S\;\_\;{Thev}} + Z} \right)}^{- H}{Z\left( {Z_{S\;\_\;{Thev}} + Z} \right)}^{- 1}v_{s}} +} \right.}} \\\left. \left( {{v_{S}^{H}\left( {Z_{S\;\_\;{Thev}} + Z} \right)}^{- H}{Z\left( {Z_{S\;\_\;{Thev}} + Z} \right)}^{- 1}v_{s}} \right)^{H} \right) \\{= {\frac{1}{2}\left( {{v_{S\;}^{H}\left( {Z_{S\;\_\;{Thev}} + Z} \right)}^{- H}\left( {Z + Z^{H}} \right)\left( {Z_{S\;\_\;{Thev}} + Z} \right)^{- 1}v_{S}} \right)}} \\{= {{v_{S}^{H}\left( {\left( {Z_{S\;\_\;{Thev}} + Z} \right)^{- H}{{Re}(Z)}\left( {Z_{S\;\_\;{Thev}} + Z} \right)^{- 1}} \right)}v_{s}}} \\{= {v_{S}^{H}Q_{Thev}v_{s}}}\end{matrix}$whereQ _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹.

Here we have used the fact that for passive linear networksZ=Z ^(T)and thus

$\frac{Z + Z^{H}}{2} = {\frac{Z + \left( Z^{T} \right)^{*}}{2} = {\frac{Z + Z^{*}}{2} = {{{Re}(Z)}.}}}$

For the Norton source model 400 with source currents i_(S) and sourceimpedance Z_(S) _(_) _(Nor), the antenna currents are given byi=Z ⁻¹(Z _(S) _(_) _(Nor) ⁻¹ +Z ⁻¹)⁻¹ i _(S),where Z is the impedance matrix for the array. If it is again assumedthat all of the power delivered to the array is radiated (i.e., no ohmicor other losses), the radiated power for the Norton source is given byP _(rad) _(_) _(Nor) _(_) _(circuit)(i _(S) ,Z _(S) _(_) _(Nor) ,Z)=Re(i_(S) ^(H)(Z _(S) _(_) _(Nor) ⁻¹ +Z ⁻¹)^(−H) Z ^(−H) ZZ ⁻¹(Z _(S) _(_)_(Nor) ⁻¹ +Z ⁻¹)⁻¹ i _(S)).=Re(i _(S) ^(H) Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Z(Z_(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor) i _(S))

It can be noted that if that if the Norton source is equivalent to theThevenin source in the previous section, thenZ _(S) _(_) _(Nor) i _(S) =v _(S) and Z _(S) _(_) _(Nor) =Z _(S) _(_)_(Thev)and the radiated power is given byP _(rad) _(_) _(Nor) _(_) _(circut)(i _(S) ,Z _(S) _(_) _(Nor) ,Z)=Re(i_(S) ^(H) Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Z(Z _(S)_(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor) i _(S)),=Re(v _(S) ^(H)(Z _(S) _(_) _(Thev) +Z)^(−H) Z(Z _(S) _(_) _(Thev) +Z)⁻¹v _(S))which is the same as for the Thevenin source model.

As in the case of the Thevenin source model, the expression for radiatedpower can be simplified as

$\begin{matrix}{{P_{{ra}\; d\;\_\;{Nor}\;\_\;{circuit}}\left( {{\mathbb{i}}_{s},Z_{S\;\_\;{Nor}},Z} \right)} = {\frac{1}{2}\left( {{\mathbb{i}}_{S}^{H}{Z_{S\;\_\;{Nor}}^{H}\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- H}{Z\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- 1}} \right.}} \\{{Z_{S\;\_\;{Nor}}{\mathbb{i}}_{S}} + \left( {{\mathbb{i}}_{S}^{H}{Z_{S\;\_\;{Nor}}^{H}\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- H}} \right.} \\\left. \left. {{Z\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- 1}Z_{S\;\_\;{Nor}}{\mathbb{i}}_{S}} \right)^{H} \right) \\{= {\frac{1}{2}\left( {{\mathbb{i}}_{S\;}^{H}{Z_{S\;\_\;{Nor}}^{H}\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- H}\left( {Z + Z^{H}} \right)} \right.}} \\\left. {\left( {Z_{S\;\_\;{Nor}} + Z} \right)^{- 1}Z_{S\;\_\;{Nor}}{\mathbb{i}}_{s}} \right) \\{= {{\mathbb{i}}_{S}^{H}\left( {{Z_{S\;\_\;{Nor}}^{H}\left( {Z_{S\;\_\;{Nor}} + Z} \right)}^{- H}{{Re}(Z)}\left( {Z_{S\;\_\;{Nor}} + Z} \right)^{- 1}} \right.}} \\{\left. Z_{S\;\_\;{Nor}} \right){\mathbb{i}}_{s}} \\{= {{\mathbb{i}}_{S}^{H}Q_{Nor}{\mathbb{i}}_{S}}}\end{matrix}$whereQ _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z_(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor).

For unit energy precoders with coupled antennas, it has now beenestablished that regardless of whether the precoder is implemented as avector voltage source (in a Thevenin source) or as a vector currentsource (in a Norton source), the radiated power can be expressed inquadratic form. In particular, for a Thevenin source, the radiated powercan be expressed asP _(rad) _(_) _(Thev)(v _(S) ,Z _(S) _(_) _(Thev) ,Z)=v _(S) ^(H) Q_(Thev) v _(S)whereQ _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹.

Similarly, for a Norton source, the radiated power can be expressed asP _(rad) _(_) _(Nor) _(_) _(circuit)(i _(S) ,Z _(S) _(_) _(Nor) ,Z)=i_(S) ^(H) Q _(Nor) i _(S)whereQ _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z_(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor).

The problem with both of these quadratic forms is that precoders withequal L² norm do not map to equal energy antenna patterns. Todemonstrate this point, we consider the case of two half-wavelengthdipoles separated by one-half wavelength. The impedance matrix for thisarray is given by

$Z = {\begin{bmatrix}{73 + {j \cdot 42.5}} & {{- 13} - {j \cdot 25}} \\{{- 13} - {j \cdot 25}} & {73 + {j \cdot 42.5}}\end{bmatrix}.}$

FIG. 5 is an example graph 500 illustrating transmitted power vs.relative phase offset for a Thevenin source model according to apossible embodiment. Consider the Thevenin source with a precoder of theform v(θ)=[1 exp(jθ)]^(T), and note that the L² norm of the precoderv(θ) is independent of the phase θ, so that∥v(θ)∥²=2for all θ. The radiated power for this precoder is shown in the graph500 as a function of the relative phase θ and the source impedance Z_(S)_(_) _(Thev), for source impedances of 0, 25, 50 and 73 −j42 ohms. Itcan be observed that for a source impedance of 0, the transmitted powervaries by 3.7 dB even as the L² norm of the precoder is held constant.

FIG. 6 is an example graph 600 illustrating transmitter power vs.relative phase offset for a Norton source model according to a possibleembodiment. Consider the Norton source with a precoder of the formi(θ)=[1 exp (jθ)]^(T), and note that the L² norm of the precoder i(θ) isindependent of the phase θ, so that∥i(θ)∥²=2for all θ. The radiated power for this precoder is shown in the graph600 as a function of the relative phase θ and the source impedance Z_(S)_(_) _(Thev), for source impedances of 100, 200, ∞, and 73 −j42 ohms. Itcan be observed that for infinite source impedance (∞), the transmittedpower varies by 1.5 dB even as the L² norm of the precoder is heldconstant.

So, in summary, even if the L² norm of the precoder is held constant,the transmitted power can vary by several dB, and in particular, canvary by 3.7 dB for a Thevenin source with zero impedance and can vary by1.5 dB for a Norton source with infinite source impedance.

As noted previously, there is a need for the system to have theprecoders map to equal energy antenna patterns. All of the standardshave already defined precoders with equal L² norms, so it is highlydesirable to find a way to use the antenna precoders which have alreadybeen defined. Furthermore, it is simply not practical to define a newset of antenna precoders for each possible antenna array configuration(and associated coupling and impedance parameters), and source model(Thevenin or Norton, with associated source impedance).

The transmitted power can vary quite significantly over the codebook.Embodiments provide a solution to the problem of transmitter powervariation over the precoder codebook by transforming the precoders totransformed precoders which map to antenna patterns with equal radiatedpower, and which does not require additional signaling to inform thereceiver of the precoder transformation used at the transmitter.Embodiments further provide a method for mapping the existing precoderswhich have already been defined in the standards (IEEE 802.11 and802.16, and 3GPP HSPA and LTE standards) into equal energy precoders.The proposed method has multiple advantages. One advantage is that theproposed method can use existing precoders. Another advantage is thatthe proposed method still allows precoder based channel estimation andchannel quality evaluation. Another advantage is that the proposedmethod does not require signaling of the precoder transformation used atthe transmitter.

In order to describe the method, we first consider the case of theThevenin source for which the radiated power is given byP _(rad) _(_) _(Thev)(v _(S) ,Z _(S) _(_) _(Thev) ,Z)=v _(S) ^(H) Q_(Thev) v _(S).Since the M×M matrix Q_(Thev) is positive definite, it can be expressedasQ _(Thev) =P _(Thev) ^(H) P _(Thev)where we refer to P_(Thev) ^(H) and P_(Thev) as the left and rightfactors of the product, where this factorization is non-unique. TheCholesky decomposition is one possible factorization of this form. Forthe Cholesky factorization, the matrix P_(Thev) is upper triangular.Other factorizations having this form can be generated by noting thatbecause Q_(Thev) is Hermitian, the eigendecomposition of Q_(Thev) hasthe formQ _(Thev) =UΛU ^(H)where the columns of U are the eigenvectors of Q_(Thev) and the matrix Λis diagonal. The diagonal elements of Λ are the eigenvaluescorresponding to the eigenvectors of Q_(Thev), where the eigenvalues inΛ are in the same order as the corresponding eigenvectors in U. Usingthis eigendecomposition, we can defineP _(Thev)=Λ^(1/2) U ^(H)where Λ^(1/2) is the square root of the matrix Λ. It can be noted thatthe eigendecomposition of the matrix Q_(Thev) is not unique since theeigenvectors forming the columns of U can be placed in any order. If thedimension of Q_(Thev) is M×M, then Q_(Thev) has M eigenvectors and thereare M factorial (M!=M*(M−1)*(M−2)* . . . *1) possible orderings of theseeigenvectors. Also, given the matrix U and the diagonal matrix of thecorresponding eigenvalues Λ, the square root matrix Λ^(1/2) isnon-unique since each eigenvalue has both a positive and a negativesquare root (all of the eigenvalues of Q_(Thev) are non-negative). Thus,given the matrix Λ, there are 2^(M) possible matrices Λ^(1/2). However,given the matrix Λ, there is only one matrix Λ^(1/2) for which all ofthe values are non-negative and we refer to this as the positive squareroot.

For the remainder of this section, we use the definitionP _(Thev)=Λ^(1/2) U ^(H)where Λ^(1/2) is the positive square root of the matrix Λ. For ourpurposes, the ordering of the eigenvectors of Q_(Thev) within thecolumns of U do not matter, though the ordering of the eigenvalues in Λmust correspond to the ordering of the eigenvectors in U. It can benoted that because the eigenvectors are orthonormal, it follows thatP _(Thev) ⁻¹ =UΛ ^(−1/2).Now definev _(S) =P _(Thev) ⁻¹ w=UΛ ^(−1/2) wso that v_(S) is the sum of the projections of w onto the eigenvectorsof Q_(Thev) scaled by the inverse square root of the correspondingeigenvalues. Note thatv _(S) ^(H) Q _(Thev) v _(S) =w ^(H) P _(Thev) ^(−H) Q _(Thev) P _(Thev)⁻¹ w=w ^(H) P _(Thev) ^(−H) P _(Thev) ^(H) P _(Thev) P _(Thev) ⁻¹ w.=w ^(H) w=∥w∥ ₂ ²Thus, if each precoder w is transformed into a voltage vector v_(S)using the transformation v_(S)=P_(Thev) ⁻¹w, all of the precoders willmap to equal energy patterns so long as all of the precoders have thesame L² norm.

In order for these precoders to be used for precoder-based channelestimation and channel quality evaluation at the receiver, the antennapatterns that must be linear with respect to the precoders. Thus, if afirst precoder w₁ produces antenna pattern q_(w) ₁ (θ,φ) and precoder w₂produces antenna pattern q_(w) ₂ (θ,φ), then it must be true that theprecoder αw₁+βw₂ produces the antenna pattern αq_(w) ₁ (θ,φ)+βq_(w) ₂(θ,φ), where α and β are complex scalar constants. Thus, so long as thereference symbol precoders, both cell-specific reference symbols (CRS)and channel state information reference symbols (CSI-RS), use the sametransformation as the data symbol precoders, the receiver can use theexisting precoders to estimate the channel, and the receiver does notneed to know the precoder transformation that was used at thetransmitter.

For the Thevenin source model, the antenna pattern resulting from theapplication of precoder w₁ is given byq _(w) ₁ (θ,φ)=v₁ ^(T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)=w ₁ ^(T) P _(Thev) ^(−T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)where p(θ,φ) is the vector of antenna element patterns in isolation fromeach other. Similarly, the antenna pattern resulting from precoder w₂ isgiven byq _(w) ₂ (θ,φ)=v₂ ^(T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ).=w ₂ ^(T) P _(Thev) ^(−T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)Finally, the antenna pattern resulting from precoder αw₁+βw₂ is given byq _(αw) ₁ _(+βw) ₂ (θ,φ)=(αv ₁ +βv ₂)^(T)(Z _(S) _(_) _(Thev) +Z)⁻¹p(θ,φ)=(αw ₁ ^(T) +βw ₂ ^(T))P _(Thev) ^(−T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)=αw ₁ ^(T) P _(Thev) ^(−T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)+βw ₂ ^(T) P_(Thev) ^(−T)(Z _(S) _(_) _(Thev) +Z)⁻¹ p(θ,φ)=αq _(w) ₁ (θ,φ)+βq _(w) ₂ (θ,φ)and thus the precoders have the linearity property needed forprecoder-based channel evaluation at the receiver.

We now consider the Norton source for which the radiated power is givenbyP _(rad) _(_) _(Nor) _(_) _(circuit)(i _(S) ,Z _(S) _(_) _(Nor) ,Z)=i_(S) ^(H) Q _(Nor) i _(S).Since the M×M matrix Q_(Nor) is positive definite, it can be expressedasQ _(Nor) =P _(Nor) ^(H) P _(Nor)where we refer P_(Nor) ^(H) and P_(Nor) as the left and right factors ofthe product, where this factorization is non-unique. As in the previoussection, the Cholesky decomposition is one factorization having thisform in which P_(Nor) is upper triangular. Also as before, otherfactorizations having this form can be generated by noting that sinceQ_(Nor) is Hermitian, the eigendecomposition of Q_(Nor) has the formQ _(Nor) =UΛU ^(H)where the columns of U are the eigenvectors of Q_(Nor) and the matrix Λis diagonal. The diagonal elements of Λ are the eigenvaluescorresponding to the eigenvectors of Q_(Nor), where the eigenvalues in Λare in the same order as the corresponding eigenvectors in U. Using theeigendecomposition, we can defineP _(Nor)=Λ^(1/2) U ^(H).where Λ^(1/2) is the square root of the matrix Λ. As in the previoussection, Λ^(1/2) is non-unique since each eigenvalue has both a positiveand a negative square root (all of the eigenvalues of Q_(Nor) arenon-negative). We refer to the single matrix Λ^(1/2) for which all ofthe values are non-negative as the positive square root.

For the remainder of this section, we use the definitionP _(Nor)=Λ^(1/2) U ^(H),where Λ^(1/2) is the positive square root of the matrix Λ. For ourpurposes, the ordering of the eigenvectors of Q_(Nor) within the columnsof U do not matter, though the ordering of the eigenvalues in Λ mustcorrespond to the ordering of the eigenvectors in U. It can be notedthat because the eigenvectors are orthonormal, it follows thatP _(Nor) ⁻¹ =UΛ ^(−1/2).

Now definei _(S) =P _(Nor) ⁻¹ w=UΛ ^(−1/2) wso that i_(S) is the sum of the projections of w onto the eigenvectorsof Q_(Nor) scaled by the inverse square root of the correspondingeigenvalues. Note thati _(S) ^(H) Q _(Nor) i _(S) =w ^(H) P _(Nor) ^(−H) Q _(Nor) P _(Nor) ⁻¹w=w ^(H) P _(Nor) ^(−H) P _(Nor) ^(H) P _(Nor) P _(Nor) ⁻¹ w=w ^(H) w=∥w∥ ₂ ²Thus, if each precoder w is transformed into a current vector i_(S)using the transformation i_(S)=P_(Nor) ⁻¹w, all of the precoders willmap to unit energy.

In order for these precoders to be used for precoder-based channelestimation channel quality evaluation at a receiver, the antennapatterns that must be linear with respect to the precoders. Thus, if afirst precoder w₁ produces antenna pattern q_(w) ₁ (θ,φ) and precoder w₂produces antenna pattern q_(w) ₂ (θ,φ), then it must be true that theprecoder αw₁+βw₂ produces the antenna pattern αq_(w) ₁ (θ,φ)+βq_(w) ₂(θ,φ), where α and β are complex scalar constants.

For the Norton source model, the antenna pattern resulting from theapplication of precoder w₁ is given byq _(w) ₁ (θ,φ)=i ₁ ^(T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)=w ₁ ^(T) P _(Nor) ^(−T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)where p(θ,φ) is the vector of antenna element patterns in isolation fromeach other. Similarly, the antenna pattern resulting from precoder w₂ isgiven byq _(w) ₂ (θ,φ)=i ₂ ^(T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ).=w ₂ ^(T) P _(Nor) ^(−T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)

Finally, the antenna pattern resulting from precoder αw₁+βw₂ is given byq _(αw) ₁ _(+βw) ₂ (θ,φ)=(αi ₁ +βi ₂)^(T)(Z _(S) _(_) _(Nor) +Z)⁻¹p(θ,φ)=(αw ₁ ^(T) +βw ₂ ^(T))P _(Nor) ^(−T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)=αw ₁ ^(T) P _(Nor) ^(−T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)+βw ₂ ^(T) P_(Nor) ^(−T)(Z _(S) _(_) _(Nor) +Z)⁻¹ p(θ,φ)=αq _(w) ₁ (θ,φ)+βq _(w) ₂ (θ,φ)and thus the precoders have the linearity property needed forprecoder-based channel evaluation at the receiver.

Thus, so long as the reference symbol precoders use the sametransformation as the data symbol precoders, the receiver can use theexisting precoders to estimate the channel, and the receiver does notneed to know the precoder transformation that was used at thetransmitter.

FIG. 7 is an example flowchart 700 illustrating the operation of awireless communication device, such as the transmitting device 110,according to a possible embodiment. The method of the flowchart 700 canbe performed in a UE, in an eNB, or in any other device that usesprecoders and has a transmitter. At 710, the flowchart 700 can begin.

At 720, a plurality of precoders can be received from a codebook in atransmitter having an antenna array. The antennas of the array can bemutually coupled in that voltage or current applied to one antennaelement can induce a voltage or current on another antenna element inthe antenna array. A precoder can be a reference symbol precoder or adata symbol precoder. Also, a precoder can be a vector or a matrix. Theprecoders can be received by a transmitting device, can be received by acontroller in a transmitting device, can be received by a precoder unitin a transmitting device, can be received from a network entity, can bereceived from memory, and/or can be otherwise received at any usefulelement in a transmitting device and from any element that can provideprecoders from a codebook.

At 730, each precoder of the plurality of precoders can be transformedusing a transformation that maps each precoder to a transformed precodersuch the transformed precoders generate antenna patterns with equalradiated power. Both data symbol precoders and reference symbolprecoders can be transformed by the same transformation. Thetransformation can map each precoder to a transformed precoder such thetransformed precoders generate antenna patterns with equal radiatedpower from the antenna array. A transformation matrix for thetransformation can be a matrix with column vectors equal to theeigenvectors of a Hermitian and non-negative definite matrix multipliedby a diagonal matrix with the value of each diagonal element equal tothe inverse of the positive square root of the eigenvalue of thecorresponding eigenvector. According to a possible embodiment, theHermitian and non-negative definite matrix can be a function of a sourcemodel of a transmitter of the transmitting device, a source impedance ofthe transmitter, an impedance matrix of the antenna array, and/or otherinformation.

According to a possible implementation, the transmitter can include atransmitter source and the transformation can includev=P _(Thev) ⁻¹ wwhere w can be the precoder from the codebook, v can be the transformedprecoder, and P_(Thev) can be based onQ _(Thev) =P _(Thev) ^(H) P _(Thev)whereQ _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹,where Z_(S) _(_) _(Thev) can be a diagonal matrix of the transmittersource impedances and Z is the impedance matrix of the antenna array.

According to another possible implementation, the transmitter caninclude a transmitter source and the transformation can includei=P _(Nor) ⁻¹ wwhere w can be the precoder from the codebook, i can be the transformedprecoder, and P_(Nor) can be based onQ _(Nor) =P _(Nor) ^(H) P _(Nor)whereQ _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z_(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor),where Z_(S) _(_) _(Nor) can be a diagonal matrix of the transmittersource impedances and Z can be the impedance matrix of the antennaarray.

At 740, a signal can be received for transmission. The signal can be areference symbol or data symbol. At 750, a transformed precoder of theplurality of transformed precoders can be applied to the signal togenerate a precoded signal for transmission over a physical channel.Applying can include multiplying the transformed precoder of theplurality of transformed precoders by the signal to generate a precodedsignal for transmission over the physical channel. At 760, the precodedsignal can be transmitted. At 770, the flowchart 700 can end.

Some example embodiments above describe a two-port model for atwo-element antenna array, and more generally, an M-port model for anM-element antenna array. In these example embodiments, impedanceparameters (Z matrix) model the relationship of the voltages and thecurrents for this two-port, or M-port, antenna array asv=Zi.

There are also other equivalent parameters that can be used to model theantenna array. For example, other sets of parameters can includeadmittance parameters (Y), hybrid parameters (H), inverse hybridparameters (G), ABCD parameters (ABCD), scattering parameters (S),scattering transfer parameters (T), and other parameters useful formodeling an antenna array. All of these models are equivalent, even ifthey look slightly different. For example, the admittance parameters (Y)can have the voltage-current relationship ofi=Yvso thatv=Y ⁻¹ iand thusZ=Y ⁻¹.

The last expression can give the relationship between the impedanceparameters and the admittance parameters. Thus, if the antenna array isrepresented in terms of its admittance parameters, the precodertransformation may look slightly different, but is still equivalent tothe transformation with the impedance parameters. For example, thetransformation can be exactly the same with the exact same or similarmapping of a precoder to a transformed precoder, except that Z can bereplaced everywhere by Y⁻¹, and these can be exactly equal. Similarly,all the other parameters cases above can be converted to Z parametersand are thus equivalent.

It should be understood that, notwithstanding the particular steps asshown in the figures, a variety of additional or different steps can beperformed depending upon the embodiment, and one or more of theparticular steps can be rearranged, repeated or eliminated entirelydepending upon the embodiment. Also, some of the steps performed can berepeated on an ongoing or continuous basis simultaneously while othersteps are performed. Furthermore, different steps can be performed bydifferent elements or in a single element of the disclosed embodiments.

FIG. 8 is an example block diagram of an apparatus 800, such as thetransmitting device 110, according to a possible embodiment. Theapparatus 800 can be a base station, a UE, or any other transmittingapparatus. The apparatus 800 can include a housing 810, a controller 820within the housing 810, audio input and output circuitry 830 coupled tothe controller 820, a display 840 coupled to the controller 820, atransceiver 850 coupled to the controller 820, an antenna arrayincluding plurality of antennas, such as antennas 855 and 857, coupledto the transceiver 850, a user interface 860 coupled to the controller820, a memory 870 coupled to the controller 820, and a network interface880 coupled to the controller 820. The apparatus 800 can perform themethods described in all the embodiments.

The display 840 can be a viewfinder, a liquid crystal display (LCD), alight emitting diode (LED) display, a plasma display, a projectiondisplay, a touch screen, or any other device that displays information.The transceiver 850 can include a transmitter and/or a receiver. Theplurality of antennas 855 and 857 can include two or more antennas. Theantennas 855 and 857 can be mutually coupled in that voltage or currentapplied to one antenna element induces a voltage or current on anotherantenna element in the antenna array. The audio input and outputcircuitry 830 can include a microphone, a speaker, a transducer, or anyother audio input and output circuitry. The user interface 860 caninclude a keypad, a keyboard, buttons, a touch pad, a joystick, a touchscreen display, another additional display, or any other device usefulfor providing an interface between a user and an electronic device. Thenetwork interface 880 can be a Universal Serial Bus (USB) port, anEthernet port, an infrared transmitter/receiver, an IEEE 1398 port, aWLAN transceiver, or any other interface that can connect an apparatusto a network, device, or computer and that can transmit and receive datacommunication signals. The memory 870 can include a random accessmemory, a read only memory, an optical memory, a flash memory, aremovable memory, a hard drive, a cache, or any other memory that can becoupled to a wireless communication device.

The apparatus 800 or the controller 820 may implement any operatingsystem, such as Microsoft Windows®, UNIX®, or LINUX®, Android™, or anyother operating system. Apparatus operation software may be written inany programming language, such as C, C++, Java or Visual Basic, forexample. Apparatus software may also run on an application framework,such as, for example, a Java® framework, a .NET® framework, or any otherapplication framework. The software and/or the operating system may bestored in the memory 870 or elsewhere on the apparatus 800. Theapparatus 800 or the controller 820 may also use hardware to implementdisclosed operations. For example, the controller 820 may be anyprogrammable processor. Disclosed embodiments may also be implemented ona general-purpose or a special purpose computer, a programmedmicroprocessor or microprocessor, peripheral integrated circuitelements, an application-specific integrated circuit or other integratedcircuits, hardware/electronic logic circuits, such as a discrete elementcircuit, a programmable logic device, such as a programmable logicarray, field programmable gate-array, or the like. In general, thecontroller 820 may be any controller or processor device or devicescapable of operating a wireless communication device and implementingthe disclosed embodiments. While the controller 820 is illustrated asone block and operations of the controller 820 can be performed in oneelement, the controller 820 can alternately be distributed betweendifferent elements of the apparatus 800 as well as distributed throughcloud computing.

In operation, the memory 870 can store a codebook including a pluralityof precoders. The controller 820 can receive a plurality of precodersfrom the codebook in the memory 870. The controller 820 can transformeach precoder of the plurality of precoders using a transformation thatmaps each precoder to a transformed precoder such the transformedprecoders generate antenna patterns with equal radiated power. Thetransformation can map each precoder to a transformed precoder such thetransformed precoders generate antenna patterns with radiated power fromthe antenna array. The same transformation can be applied to data symbolprecoders and reference symbol precoders. A transformation matrix forthe transformation can be a matrix with column vectors equal to theeigenvectors of a Hermitian and non-negative definite matrix multipliedby a diagonal matrix with the value of each diagonal element equal tothe inverse of the positive square root of the eigenvalue of thecorresponding eigenvector. Furthermore, the Hermitian and non-negativedefinite matrix can be a function of a source model of the transmitter,a source impedance of the transmitter, an impedance matrix of theantenna array, and/or other information.

According to a possible embodiment, the transceiver 850 can include atransmitter source 852 and the transformation can includev=P _(Thev) ⁻¹ wwhere w can be the precoder from the codebook, v can be the transformedprecoder, and P_(Thev) can be based onQ _(Thev) =P _(Thev) ^(H) P _(Thev)whereQ _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹where Z_(S) _(_) _(Thev) can be a diagonal matrix of the transmittersource impedances and Z can be the impedance matrix of the antennaarray.

According to another possible embodiment, the transformation can includei=P _(Nor) ⁻¹ wwhere w can be the precoder from the codebook, i is the transformedprecoder, and P_(Nor) can be based onQ _(Nor) =P _(Nor) ^(H) P _(Nor)whereQ _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z_(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor)where Z_(S) _(_) _(Nor) is a diagonal matrix of the transmitter sourceimpedances and Z is the impedance matrix of the antenna array.

The controller 820 can receive a signal for transmission. The signal canbe a reference signal or a data signal. The controller 820 can apply atransformed precoder of the plurality of transformed precoders to thesignal to generate a precoded signal for transmission over a physicalchannel. The controller 820 can apply a transformed precoder bymultiplying the transformed precoder by the signal to generate aprecoded signal for transmission over a physical channel. Thetransceiver 850 can transmit the precoded signal.

The method of this disclosure can be implemented on a programmedprocessor. However, the controllers, flowcharts, and modules may also beimplemented on a general purpose or special purpose computer, aprogrammed microprocessor or microcontroller and peripheral integratedcircuit elements, an integrated circuit, a hardware electronic or logiccircuit such as a discrete element circuit, a programmable logic device,or the like. In general, any device on which resides a finite statemachine capable of implementing the flowcharts shown in the figures maybe used to implement the processor functions of this disclosure.

While this disclosure has been described with specific embodimentsthereof, it is evident that many alternatives, modifications, andvariations will be apparent to those skilled in the art. For example,various components of the embodiments may be interchanged, added, orsubstituted in the other embodiments. Also, all of the elements of eachfigure are not necessary for operation of the disclosed embodiments. Forexample, one of ordinary skill in the art of the disclosed embodimentswould be enabled to make and use the teachings of the disclosure bysimply employing the elements of the independent claims. Accordingly,embodiments of the disclosure as set forth herein are intended to beillustrative, not limiting. Various changes may be made withoutdeparting from the spirit and scope of the disclosure.

In this document, relational terms such as “first,” “second,” and thelike may be used solely to distinguish one entity or action from anotherentity or action without necessarily requiring or implying any actualsuch relationship or order between such entities or actions. The phrase“at least one of” followed by a list is defined to mean one, some, orall, but not necessarily all of, the elements in the list. The terms“comprises,” “comprising,” or any other variation thereof, are intendedto cover a non-exclusive inclusion, such that a process, method,article, or apparatus that comprises a list of elements does not includeonly those elements but may include other elements not expressly listedor inherent to such process, method, article, or apparatus. An elementproceeded by “a,” “an,” or the like does not, without more constraints,preclude the existence of additional identical elements in the process,method, article, or apparatus that comprises the element. Also, the term“another” is defined as at least a second or more. The terms“including,” “having,” and the like, as used herein, are defined as“comprising.” Furthermore, the background section is written as theinventor's own understanding of the context of some embodiments at thetime of filing and includes the inventor's own recognition of anyproblems with existing technologies and/or problems experienced in theinventor's own work.

We claim:
 1. A method comprising: receiving a plurality of precoders from a codebook in a transmitter having an antenna array; transforming each precoder of the plurality of precoders using a transformation that maps each precoder to a transformed precoder such the transformed precoders generate antenna patterns with equal radiated power, where a transformation matrix for the transformation is a matrix with column vectors equal to the eigenvectors of a Hermitian and non-negative definite matrix multiplied by a diagonal matrix with the value of each diagonal element equal to the inverse of the positive square root of the eigenvalue of the corresponding eigenvector; receiving a signal for transmission; applying a transformed precoder of the plurality of transformed precoders to the signal to generate a precoded signal for transmission over a physical channel; and transmitting the precoded signal.
 2. The method according to claim 1, wherein the transformation maps each precoder to a transformed precoder such the transformed precoders generate antenna patterns with equal radiated power from the antenna array.
 3. The method according to claim 1, wherein the Hermitian and non-negative definite matrix is a function of a source model of the transmitter, a source impedance of the transmitter, and an impedance matrix of the antenna array.
 4. The method according to claim 3, wherein the transmitter includes a transmitter source, and wherein the transformation comprises: v=P _(Thev) ⁻¹ w where w is the precoder from the codebook, v is the transformed precoder, and P_(Thev) is based on Q _(Thev) =P _(Thev) ^(H) P _(Thev) where Q _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹ where Z_(S) _(_) _(Thev) is a diagonal matrix of the transmitter source impedances and Z is the impedance matrix of the antenna array.
 5. The method according to claim 3, wherein the transmitter includes a transmitter source, and wherein the transformation comprises: i=P _(Nor) ⁻¹ w where w is the precoder from the codebook, i is the transformed precoder, and P_(Nor) is based on Q _(Nor) =P _(Nor) ^(H) P _(Nor) where Q _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor) where Z_(S) _(_) _(Nor) is a diagonal matrix of the transmitter source impedances and Z is the impedance matrix of the antenna array.
 6. The method according to claim 1, wherein applying comprises multiplying the transformed precoder of the plurality of transformed precoders by the signal to generate a precoded signal for transmission over a physical channel.
 7. The method according to claim 1, wherein both data symbol precoders and reference symbol precoders are transformed by the same transformation.
 8. The method according to claim 1, wherein the antennas of the array are mutually coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the antenna array.
 9. An apparatus comprising: an antenna array; a transceiver coupled to the antenna array; a memory to store a codebook including a plurality of precoders; a controller coupled to the transceiver and the memory, the controller to receive a plurality of precoders from the codebook, transform each precoder of the plurality of precoders using a transformation that maps each precoder to a transformed precoder such the transformed precoders generate antenna patterns equal radiated power, where a transformation matrix for the transformation is a matrix with column vectors equal to the eigenvectors of a Hermitian and non-negative definite matrix multiplied by a diagonal matrix with the value of each diagonal element equal to the inverse of the positive square root of the eigenvalue of the corresponding eigenvector, receive a signal for transmission, and apply a transformed precoder of the plurality of transformed precoders to the signal to generate a precoded signal for transmission over a physical channel, wherein the transceiver transmits the precoded signal.
 10. The apparatus according to claim 9, wherein the transformation maps each precoder to a transformed precoder such the transformed precoders generate antenna patterns with equal radiated power from the antenna array.
 11. The apparatus according to claim 9, wherein the Hermitian and non-negative definite matrix is a function of a source model of the transceiver, a source impedance of the transceiver, and an impedance matrix of the antenna array.
 12. The apparatus according to claim 11, wherein the transceiver includes a transmitter source, and wherein the transformation comprises: v=P _(Thev) ⁻¹ w where w is the precoder from the codebook, v is the transformed precoder, and P_(Thev) is based on Q _(Thev) =P _(Thev) ^(H) P _(Thev) where Q _(Thev)=(Z _(S) _(_) _(Thev) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Thev) +Z)⁻¹ where Z_(S) _(_) _(Thev) is a diagonal matrix of the transmitter source impedances and Z is the impedance matrix of the antenna array.
 13. The apparatus according to claim 11, wherein the transceiver includes a transmitter source, and wherein the transformation comprises: i=P _(Nor) ⁻¹ w where w is the precoder from the codebook, i is the transformed precoder, and P_(Nor) is based on Q _(Nor) =P _(Nor) ^(H) P _(Nor) where Q _(Nor) =Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z)^(−H) Re(Z)(Z _(S) _(_) _(Nor) +Z)⁻¹ Z _(S) _(_) _(Nor) where Z_(S) _(_) _(Nor) is a diagonal matrix of the transmitter source impedances and Z is the impedance matrix of the antenna array.
 14. The apparatus according to claim 9, wherein the same transformation is applied to data symbol precoders and reference symbol precoders.
 15. The apparatus according to claim 9, wherein the controller applies a transformed precoder by multiplying the transformed precoder by the signal to generate a precoded signal for transmission over a physical channel.
 16. The apparatus according to claim 9, wherein the antennas of the array are mutual coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the antenna array.
 17. A method comprising: receiving a plurality of precoders from a codebook in a transmitter having an antenna array where the antennas of the array are mutual coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the antenna array; transforming each precoder of the plurality of precoders using a transformation that maps each precoder to a transformed precoder such the transformed precoders generate antenna patterns with equal radiated power from the antenna array, where a transformation matrix for the transformation is a matrix with column vectors equal to the eigenvectors of a Hermitian and non-negative definite matrix multiplied by a diagonal matrix with the value of each diagonal element equal to the inverse of the positive square root of the eigenvalue of the corresponding eigenvector; receiving a signal for transmission; multiplying the transformed precoder of the plurality of transformed precoders by the signal to generate a precoded signal for transmission over a physical channel; and transmitting the precoded signal, wherein the same transformation is applied to data symbol precoders and reference symbol precoders. 